Answer:
Explanation:
A. To find the formula for f'(x) using the limit definition of the derivative, we need to evaluate the following limit:
f'(x) = lim(h->0) [f(x + h) - f(x)] / h
Substituting f(x) = 3x^2 - 5, we get:
f'(x) = lim(h->0) [3(x + h)^2 - 5 - (3x^2 - 5)] / h
Expanding the square, simplifying, and canceling out the constant terms, we get:
f'(x) = lim(h->0) [6xh + 3h^2] / h
Canceling out the h's, we get:
f'(x) = lim(h->0) (6x + 3h)
Taking the limit as h approaches 0, we get:
f'(x) = 6x
Therefore, the formula for f'(x) is:
f'(x) = 6x
B. To evaluate f'(-1), we simply substitute x = -1 into the formula we found in part A:
f'(-1) = 6(-1) = -6
Therefore, f'(-1) = -6.